limits of x are 0 and +h. We thus get for the
disturbance at [xi], [eta], due to this stream

2lh f k[eta]l 2f k[xi]h / [xi]h \
- --------- . ------- sin ------- . ------ sin ------ . sin k (at - f - R + ----- ) (2).
[lambda]f k[eta]l f k[xi]h 2f \ 2f /

If we put for shortness [pi] for the quantity under the last circular
function in (1), the expressions (1), (2) may be put under the forms u
sin [tau], v sin ([tau] - [alpha]) respectively; and, if I be the
intensity, I will be measured by the sum of the squares of the
coefficients of sin [tau] and cos [tau] in the expression

u sin[tau] + v sin([tau] - [alpha]),

so that

I = u^2 + v^2 + 2uv cos[alpha],

which becomes on putting for u, v, and [alpha] their values, and
putting

/ f k[eta]l \^2
( ------- sin ------- ) = Q (3),
\k[eta]l f /
_ _
4l^2 [pi][xi]h | / 2[pi]R 2[pi][xi]h\ |
I = Q . ------------ sin^2 --------- |2 + 2 cos ( -------- - ---------- ) | (4).
[pi]^2[xi]^2 [lambda]f |_ \[lambda] [lambda]f / _|

If the subject of examination be a luminous line parallel to [eta], we
shall obtain what we require by integrating (4) with respect to [eta]
from -[oo] to +[oo]. The constant multiplier is of no especial
interest so that we may take as applicable to the image of a line
_ _
2 [pi][xi]h | / 2[pi]R 2[pi][xi]h \ |
I = ------ sin^2 --------- |1 + cos ( -------- - ---------- ) | (5).
[xi]^2 [lambda]f |_ \[lambda] [lambda]f / _|

If R = 1/2[lambda], I vanishes at [xi]= 0; but the whole illumination,
represented by
_
/ +[oo]
| I d[xi], is independent of the value of R. If R = 0,
_/-[oo]

1 2[pi][xi]h
I = ------ sin^2 ----------,
[xi]^2 [lambda]f

in agreement with S 3, where a has the meaning here attached to 2h.

The expression (5) gives the illumination at [xi] due to that part of
the complete image whose geometrical focus is at [xi] = 0, the
retardation for this component being R. Since we have now to integrate
for the whole il